Transcript Slide 1

s
A
r
2b
2a
sA
Topics
Fracture Mechanics-Brittle Fracture
 Fracture at Atomic Level
 Solid / Vapour Interfaces
Boundaries in Single Phase Solids
Interphase Interfaces in Solids
Interface Migration
1
Fracture Mechanics-Brittle fracture
Fracture mechanics is used to formulate quantitatively
•
The degree of Safety of a structure against brittle fracture
•
The conditions necessary for crack initiation, propagation and
arrest
•
The residual life in a component subjected to dynamic/fatigue
loading
Fracture Mechanics-Brittle fracture
Fracture mechanics identifies three primary factors that control the
susceptibility of a structure to brittle failure.
1. Material Fracture Toughness. Material fracture toughness may be
defined as the ability to carry loads or deform plastically in the
presence of a notch. It may be described in terms of the critical
stress intensity factor, KIc, under a variety of conditions. (These
terms and conditions are fully discussed in the following chapters.)
2. Crack Size. Fractures initiate from discontinuities that can vary from
extremely small cracks to much larger weld or fatigue cracks.
Furthermore, although good fabrication practice and inspection
can minimize the size and number of cracks, most complex
mechanical
components
cannot
be
fabricated
without
discontinuities of one type or another.
3. Stress Level. For the most part, tensile stresses are necessary for
brittle fracture to occur. These stresses are determined by a
stress analysis of the particular component.
Other factors such as temperature, loading rate, stress
concentrations, residual stresses, etc., influence these three
primary factors.
Fracture at the Atomic level
Repulsion
Potential Energy
 Two atoms or a set of atoms are bonded together
by cohesive energy or bond energy.
 Two atoms (or sets of atoms) are said to be
fractured if the bonds between the two atoms (or
sets of atoms) are broken by externally applied
tensile load
Distance
Bond
Energy
Attraction
Equilibrium
Distance xo
Tension
If a tensile force ‘T’ is applied to separate the two
atoms, then bond or cohesive energy is

   T dx
Bond
Energy
Applied Force
Theoretical Cohesive Stress

k
Compression
(2.1)
is the equilibrium spacing between two
xo
Where x
o
atoms.
Idealizing force-displacement relation as one half of
sine wave
  T sin ( )
C
x

(2.2)
+
+
xo
Cohesive
Force
Distance
Theoretical Cohesive Stress (Contd.)
Assuming that the origin is defined at x o and for small displacement
x
relationship is assumed to be linear such that sin( ) 
Hence force
displacement relationship is given by
x

T T
x

C
(2.2)
Bond stiffness ‘k’ is given by
k 
T
C

(2.3)
If there are n bonds acing per unit area and assuming x o as gage length and
multiplying eq. 2.3 by n then ‘k’ becomes young’s modulus and becomes
cohesive stress s C such that T C x o
sc 
Or
sc 
E
(2.4)
x o
E

(2.5)
If  is assumed to be approximately equal to the atomic spacing
Theoretical Cohesive Stress (Contd.)
The surface energy can be estimated as
 
s

1
2
s
0
C
sin 
x

 dx  s

C

(2.6)
The surface energy per unit area s equal to one half the fracture
energy because two surfaces are created when a material fractures.
Using eq. 2.4 in to eq.2.6
s 
C
E
x
o
s
(2.7)
Fracture stress for realistic material
Inglis (1913) analyzed for the flat plate with an
elliptical hole with major axis 2a and minor axis 2b,
subjected to far end stress s The stress at the tip of the
major axis (point A) is given by
2a 

sA  s 1 

b 

(2.8)
s
A
The ratio
is defined as the stress concentration
s
factor, k t
When a = b, it is a circular hole, thenk t  3.
When b is very very small, Inglis define radius of
curvature as
2
b
(2.9)
r
a
And the tip stress as
sA

a 
 s 1  a   

 r  

(2.10)
s
A
r
2b
2a
sA
Fracture stress for realistic material (contd.)
When a >> b eq. 2.10 becomes
a
s A  2s  
r
(2.11)
For a sharp crack, a >>> b, r  0 and stress at the crack tip tends to 
Assuming that for a metal, plastic deformation is zero and the sharpest crack
may have root radius as atomic spacing r  x othen the stress is given by
 a 
(2.12)
s
 2s
A


 xo 
When far end stress reaches fracture stresss  s f , crack propagates and the
stress at A reaches cohesive stress s A  s C then using eq. 2.7
(2.13)
This would
 Es 
sf  

 4a 
1/ 2
Griffith’s Energy balance approach
•First documented paper on fracture
(1920) Considered as father of
Fracture Mechanics
Griffith’s Energy balance approach (Contd.)
A A Griffith laid the foundations of modern fracture mechanics by designing a
criterion for fast fracture. He assumed that pre-existing flaws propagate under the
influence of an applied stress only if the total energy of the system is thereby
reduced. Thus, Griffith's theory is not concerned with crack tip processes or the
micromechanisms by which a crack advances.
Griffith proposed that ‘There is a simple energy
s
B
balance consisting of the decrease in potential
energy with in the stressed body due to crack
Y
extension and this decrease is balanced by increase
in surface energy due to increased crack surface’
X
2a
Griffith theory establishes theoretical strength
of brittle material and relationship between
fracture strength s f and flaw size ‘a’
s
Griffith’s Energy balance approach (Contd.)
s
B
The initial strain energy for the uncracked plate per
thickness is
2
s
(2.14)
U  
dA
i
A
Y
X
2E
2a
On creating a crack of size 2a, the tensile force on an
element ds on elliptic hole is relaxed froms  dx to
zero. The elastic strain energy released per unit width
due to introduction of a crack of length 2a is given by
a
U a   4  12 s  dx  v
w here displacem ent
0
(2.15)
u sin g x  a  cos 
s
E
s a
2
Ua 
v 
E
2
a  sin 
s
Griffith’s Energy balance approach (Contd.)
s
External work = U w   Fdy ,

(2.16)
B
w here F= resultant force = s  area
Y
 =total relative displacem ent
The potential or internal energy of the body is
X
2a
U p =U i +U a -U w
Due to creation of new surface increase in surface
energy is
(2.17)
U = 4a 

s a
Ua 
E
2 2
s

A
s
2
2E
s a
2
dA 
E
2
  Fdy  4 a  s

(2.18)
P1
Load, P
The total elastic energy of the cracked
plate is
Ut 
s
P2
Crack begins
to grow from
length (a)
Crack is
longer by an
increment (da)
)
(a
)
da
+
(a
v
Displacement, v
Griffith’s Energy balance approach (Contd.)
s
The variation of U t with crack extension
should be minimum
0 
2s a
da
E
 4s  0
Denoting s as s f during fracture
 2Es 
sf  

 a 
Energy, U
dU t
2
rf
Su
ac
e
e
En
rgy
U
=
4a
(a)
Crack
length, a
1/ 2
Total energy
(2.19)
for plane stress and
Unstable
(2.20)
The Griffith theory is obeyed by materials
which fail in a completely brittle elastic
manner, e.g. glass, mica, diamond and
refractory metals.
s 2a 2
E
Potential energy  ¶U 
release rate G =  ¶a 
Rates, G,
for plane strain
Elastic Strain
energy released
Ua 
s
2Es


sf  
2 
  a (1   ) 
Stable
1/ 2
Syrface energy/unit
extension =
(b)
Crack
length, a
ac
(a) Variation of Energy with Crack length
(b) Variation of energy rates with crack length
Griffith’s Energy balance approach (Contd.)
Griffith extrapolated surface tension values of soda lime glass from high
temperature to obtain the value at room temperature as  s  0.54 J / m 2 .
1/ 2
Using value of E = 62GPa,The value of
 2Es 

 as
  
0.1 M P a m . From the
experimental study on spherical vessels he calculated
 2Es 
s ca  

  
1/ 2
as 0.25 –
0.28 M P a m .
However, it is important to note that according to the Griffith theory, it is
impossible to initiate brittle fracture unless pre-existing defects are
present, so that fracture is always considered to be propagation- (rather
than nucleation-) controlled; this is a serious short-coming of the theory.
Modification for Ductile Materials
For more ductile materials (e.g. metals and plastics) it is found that the
functional form of the Griffith relationship is still obeyed, i.e.
1/ 2
s f  a . However, the proportionality constant can be used to evaluate s
(provided E is known) and if this is done, one finds the value is many orders of
magnitude higher than what is known to be the true value of the surface
energy (which can be determined by other means). For these materials plastic
deformation accompanies crack propagation even though fracture is
macroscopically brittle; The released strain energy is then largely dissipated by
producing localized plastic flow at the crack tip. Irwin and Orowan modified
the Griffith theory and came out with an expression
Where prepresents energy expended in plastic work. Typically for cleavage in
metallic materials p=104 J/m2 and s=1 J/m2. Since p>> s we have
 2E(s  p ) 
sf  

a


1/ 2
 2E p 
sf  

 a 
1/ 2
Strain Energy Release Rate
The strain energy release rate usually referred to
G 
dU
da
Note that the strain energy release rate is respect to crack length and most
definitely not time. Fracture occurs when reaches a critical value which is
denoted G c.
At fracture we have G  G so that
1  EG c 
sf 


Y  a 
1/ 2
c
One disadvantage of usingG c is that in order to determine s f it is necessary to
know E as well as G . This can be a problem with some materials, eg polymers
c
and composites, where varies with composition and processing. In practice, it
is usually more convenient to combine E andG cin a single fracture toughness
parameterK c where K c2  E
. G c Then can be simply determined experimentally
using procedures which are well established.
LINEAR ELASTIC FRACTURE MECHANICS (LEFM)
For LEFM the structure obeys Hooke’s law and global behavior is linear and
if any local small scale crack tip plasticity is ignored
The fundamental principle of fracture mechanics is that the stress field
around a crack tip being characterized by stress intensity factor K
which is related to both the stress and the size of the flaw. The analytic
development of the stress intensity factor is described for a number of
common specimen and crack geometries below.
LINEAR ELASTIC FRACTURE MECHANICS (LEFM)
Mode I - Opening mode: where the crack surfaces separate
symmetrically with respect to the plane occupied by the crack prior to
the deformation (results from normal stresses perpendicular to the
crack plane);
Mode II - Sliding mode: where the crack surfaces glide over one
another in opposite directions but in the same plane (results from inplane shear); and
Mode III - Tearing mode: where the crack surfaces are displaced in the
crack plane and parallel to the crack front (results from out-of-plane
shear).
LINEAR ELASTIC FRACTURE MECHANICS (Contd.)
In the 1950s Irwin [7] and coworkers introduced the concept of
stress intensity factor, which defines the stress field around the crack
tip, taking into account crack length, applied stress s and shape
factor Y( which accounts for finite size of the component and local
geometric features).
The Airy stress function.
In stress analysis each point, x,y,z, of a stressed solid undergoes the
stresses; sx sy, sz, txy, txz,tyz. With reference to figure 2.3, when a
body is loaded and these loads are within the same plane, say the x-y
plane, two different loading conditions are possible:
LINEAR ELASTIC FRACTURE MECHANICS (Contd.)
1. plane stress (PSS), when
the thickness of the body is
comparable to the size of
the plastic zone and a free
contraction
of
lateral
surfaces occurs, and,
2. plane strain (PSN), when
the specimen is thick
enough
to
avoid
contraction in the thickness
z-direction.
s
Thickness
B
Thickness
B
s
y
s
syy
s
sz
sz
sz
sz
Crack
Plane
X
a
s
s
Plane Stress
s
Plane Strain
In the former case, the overall stress state is reduced to the three
components; sx, sy, txy, since; sz, txz, tyz= 0, while, in the latter case, a
normal stress, sz, is induced which prevents the z
displacement, ez = w = 0. Hence, from Hooke's law:
sz =  (sx+sy)
where  is Poisson's ratio.
For plane problems, the equilibrium conditions are:
¶s
¶x
x

¶t
xy
¶y
0 ;
¶s
¶y
y

¶t
xy
¶x
0
If  is the Airy’s stress function satisfying the biharmonic compatibility
Conditions
  0
4
¶ 
2
sx 
Then
¶y
2
¶ 
2
, sy 
¶x
2
¶ 
2
, t xy  
¶ xy
For problems with crack tip Westergaard introduced Airy’s stress
function as


  R e[ Z ]  y Im [Z ]
Where Z is an analytic complex function
bg
Z z  R e[ z ]  y Im [ z ] ; z = x + iy


nd
st
And
Z , Zare 2 and 1 integrals of Z(z)
Then the stresses are given by
¶ 
2
sx 
¶y
 R e[ Z ]  y Im [ Z ]
'
2
¶ 
2
sy 
¶x
 R e[ Z ]  y Im [ Z ]
'
2
¶ 
2
t xy 
  y Im [ Z ]
'
¶ xy
'
w here Z = dZ dz
Opening mode analysis or Mode I
Consider an infinite plate a crack of length 2a subjected to a biaxial
State of stress. Defining:
Z 
s z
z
2
a
s
2

Boundary Conditions :
• At infinity
 | z |   s x  s y  s , t xy  0
• On crack faces
s
y
x
2a
  a  x  a ; y  0  s x  t xy  0
s
By replacing z by z+a , origin shifted to crack tip.
Z 
b g
zb
z  2a g
s za
And when |z|0 at the vicinity of the crack tip
Z 
sa

2 az
KI  s
KI
2 z
a
KI must be real and a constant at the crack tip. This is due to a
Singularity given by 1
z
The parameter KI is called the stress
intensity factor for opening
mode I.
Since origin is shifted to crack tip, it is
easier to use polar Coordinates, Using
ze
i
Further Simplification gives:


 3  
cos   1  sin   sin 


2
2
2
2r
 
 


sx 
KI
sy 
KI
t xy 
KI
In general s ij 


 3  
cos   1  sin   sin 

2r
 2  
2
 2  


 3  
sin   cos   cos 

2r
 2  
2
 2  
KI
2r
f ij    and K I  Y s  a
w here Y = configuration factor
From Hooke’s law, displacement field can be obtained as
u 
2 (1   )
E
v 
2 (1   )
E
w here
KI
   1
2   
cos  
 sin  
2
 2   2
 2  
KI
  1
2   
sin  
 cos  
2
 2   2
 2  
r
r
u , v = displacem ents in x, y directions
  (3  4  ) for plane stress problem s
3 
 
 for plane strain problem s
1  
The vertical displacements at any position along x-axis (  0 is given by
v 
s
E
a
2
x
s (1   )
2
v 
E
2

a
for plane stress
2
x
2

for plane strain
y
v
x
x
The strain energy required for creation of crack is given by the work done by
force acting on the crack face while relaxing the stress s to zero
Ua 
F or plane stress
a
s
0
E
U a  4 s 
s a
2
2
a
1
Fv
2
F or plane strain
2
x
2
dx
a
s (1   )
0
E
2
U a  4 s 
a
 s a (1   )
2
E
2
2
E
T he strain energy release rate is given by G  dU a da
s a
2
GI =
E
2
GI =
2
GI =
KI
E
 s (1   )a
2
E
K I (1   )
2
GI =
2
E
2
x
2
dx
Sliding mode analysis or Mode 2
For problems with crack tip under shear loading, Airy’s stress
function is taken as

 II   y R e[Z ]
Using Air’s definition for stresses
¶ 
t0
y
2
sx 
¶y
 2 Im [ Z ]  y R e[ Z ]
'
2
¶ 
2a
2
sy 
¶x
  y R e[ Z ]
'
2
¶ 
t0
2
t xy  
 R e[ Z ]  y Im [ Z ]
'
¶ xy
Using a Westergaard stress function of the form
Z 
t0 z
z
2
a
2

Boundary Conditions :
• At infinity
 | z |   s x  s y  0, t xy  t 0
• On crack faces
  a  x  a ; y  0  s x  t xy  0
With usual simplification would give the stresses as

 

 3   
cos    cos    2  cos   cos 
 
2r
2
 2 
2
 2  
sx 
K II
sy 
K II
t xy 
K II


 3  
cos   sin   cos 

2r
 2  
2
 2  


 3  
cos   1  sin   sin 


2
2
2
2r
 
 


Displacement components are given
by
u 
K II
E
v 
K II
E

(1   ) sin      2  cos    
2
2
r

(1   ) cos      2  cos    
2
2
r
K II  t o
a
2
KI
GI =
for plane stress
E
K I (1   )
2
GI =
2
for plane strain
E
Tearing mode analysis or Mode 3
In this case the crack is displaced along z-axis. Here the displacements u
and v are set to zero and hence
e x  e y   x y   yx  0
 x y   yx 
¶w
a n d  yz   z y 
¶x
¶w
¶y
th e e q u ilib riu m e q u a tio n is w ritte n a s
¶t xz
¶x

¶ t yz
¶y
 0
S tra in d isp la c e m e n t re la tio n sh ip is g iv e n b y
¶ w
2
¶x
2
¶ w
2

¶y
2
  w  0
2
if w is taken as
w 

1
Im [ Z ]
G
T h en
t x y  Im [ Z ];
t yz  R e[ Z ]
Using Westergaard stress functionas
Z 
t0 z
z
2
a
2

w h ere t 0 is th e ap p lied b ou n d ary sh ear str ess
w ith th e b ou n d ary con d ition s
on th e crack face
  a  x  a ; y  0  s z  t yz  t xy  0
on th e b ou n d ary x  y   , t yz  t 0
T he stresses are given by

sin  
2r
2
t xz 
K III
t yz 
K III

cos  
2r
2
s x  s y  t xy  0
and displacem ents are given by
w 
K III
G

sin  

2
2r
u  v 0
K III  t o
a